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Properties

Label 8.48.1.be.1
Level $8$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $8$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse and Zureick-Brown (RZB) label: X263
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.48.1.84

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}3&4\\6&5\end{bmatrix}$, $\begin{bmatrix}5&0\\0&5\end{bmatrix}$, $\begin{bmatrix}7&5\\0&5\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^3:C_4$
Contains $-I$: yes
Quadratic refinements: 16.96.1-8.be.1.1, 16.96.1-8.be.1.2, 16.96.1-8.be.1.3, 16.96.1-8.be.1.4, 16.96.1-8.be.1.5, 16.96.1-8.be.1.6, 48.96.1-8.be.1.1, 48.96.1-8.be.1.2, 48.96.1-8.be.1.3, 48.96.1-8.be.1.4, 48.96.1-8.be.1.5, 48.96.1-8.be.1.6, 80.96.1-8.be.1.1, 80.96.1-8.be.1.2, 80.96.1-8.be.1.3, 80.96.1-8.be.1.4, 80.96.1-8.be.1.5, 80.96.1-8.be.1.6, 112.96.1-8.be.1.1, 112.96.1-8.be.1.2, 112.96.1-8.be.1.3, 112.96.1-8.be.1.4, 112.96.1-8.be.1.5, 112.96.1-8.be.1.6, 176.96.1-8.be.1.1, 176.96.1-8.be.1.2, 176.96.1-8.be.1.3, 176.96.1-8.be.1.4, 176.96.1-8.be.1.5, 176.96.1-8.be.1.6, 208.96.1-8.be.1.1, 208.96.1-8.be.1.2, 208.96.1-8.be.1.3, 208.96.1-8.be.1.4, 208.96.1-8.be.1.5, 208.96.1-8.be.1.6, 240.96.1-8.be.1.1, 240.96.1-8.be.1.2, 240.96.1-8.be.1.3, 240.96.1-8.be.1.4, 240.96.1-8.be.1.5, 240.96.1-8.be.1.6, 272.96.1-8.be.1.1, 272.96.1-8.be.1.2, 272.96.1-8.be.1.3, 272.96.1-8.be.1.4, 272.96.1-8.be.1.5, 272.96.1-8.be.1.6, 304.96.1-8.be.1.1, 304.96.1-8.be.1.2, 304.96.1-8.be.1.3, 304.96.1-8.be.1.4, 304.96.1-8.be.1.5, 304.96.1-8.be.1.6
Cyclic 8-isogeny field degree: $2$
Cyclic 8-torsion field degree: $8$
Full 8-torsion field degree: $32$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x y + y z - z^{2} $
$=$ $16 x^{2} - 2 x y - y^{2} + 3 y z - 3 z^{2} + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 4 x^{3} z - 2 x^{2} y^{2} + 8 x z^{3} - 4 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^6\,\frac{619920xz^{11}+467424xz^{9}w^{2}+129600xz^{7}w^{4}+16448xz^{5}w^{6}+944xz^{3}w^{8}-109620y^{2}z^{10}-94095y^{2}z^{8}w^{2}-32292y^{2}z^{6}w^{4}-6000y^{2}z^{4}w^{6}-756y^{2}z^{2}w^{8}-65y^{2}w^{10}+593460yz^{11}+525204yz^{9}w^{2}+188568yz^{7}w^{4}+36608yz^{5}w^{6}+4300yz^{3}w^{8}+260yzw^{10}-529254z^{12}-376488z^{10}w^{2}-94788z^{8}w^{4}-10304z^{6}w^{6}-566z^{4}w^{8}-24z^{2}w^{10}+2w^{12}}{z^{4}(22960xz^{7}-6304xz^{5}w^{2}+928xz^{3}w^{4}-64xzw^{6}-4060y^{2}z^{6}+697y^{2}z^{4}w^{2}-76y^{2}z^{2}w^{4}+4y^{2}w^{6}+21980yz^{7}-3180yz^{5}w^{2}+336yz^{3}w^{4}-16yzw^{6}-19602z^{8}+6240z^{6}w^{2}-1098z^{4}w^{4}+120z^{2}w^{6}-8w^{8})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.s.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.v.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bc.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.0.bf.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
8.24.1.o.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.r.1 $8$ $2$ $2$ $1$ $0$ dimension zero
8.24.1.s.1 $8$ $2$ $2$ $1$ $0$ dimension zero

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
8.96.1.n.1 $8$ $2$ $2$ $1$ $0$ dimension zero
16.96.3.cx.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
16.96.3.cy.1 $16$ $2$ $2$ $3$ $0$ $2$
16.96.3.cz.1 $16$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.1.cw.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.vd.1 $24$ $3$ $3$ $9$ $2$ $1^{8}$
24.192.9.id.1 $24$ $4$ $4$ $9$ $0$ $1^{8}$
40.96.1.cn.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.fj.1 $40$ $5$ $5$ $17$ $5$ $1^{14}\cdot2$
40.288.17.na.1 $40$ $6$ $6$ $17$ $4$ $1^{14}\cdot2$
40.480.33.xp.1 $40$ $10$ $10$ $33$ $10$ $1^{28}\cdot2^{2}$
48.96.3.gl.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.gm.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.gn.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
56.96.1.cn.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.id.1 $56$ $8$ $8$ $25$ $4$ $1^{20}\cdot2^{2}$
56.1008.73.vd.1 $56$ $21$ $21$ $73$ $24$ $1^{16}\cdot2^{26}\cdot4$
56.1344.97.uv.1 $56$ $28$ $28$ $97$ $28$ $1^{36}\cdot2^{28}\cdot4$
80.96.3.hn.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.ho.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.hp.1 $80$ $2$ $2$ $3$ $?$ not computed
88.96.1.cn.1 $88$ $2$ $2$ $1$ $?$ dimension zero
104.96.1.cn.1 $104$ $2$ $2$ $1$ $?$ dimension zero
112.96.3.gl.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.gm.1 $112$ $2$ $2$ $3$ $?$ not computed
112.96.3.gn.1 $112$ $2$ $2$ $3$ $?$ not computed
120.96.1.qs.1 $120$ $2$ $2$ $1$ $?$ dimension zero
136.96.1.cn.1 $136$ $2$ $2$ $1$ $?$ dimension zero
152.96.1.cn.1 $152$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qs.1 $168$ $2$ $2$ $1$ $?$ dimension zero
176.96.3.gl.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.gm.1 $176$ $2$ $2$ $3$ $?$ not computed
176.96.3.gn.1 $176$ $2$ $2$ $3$ $?$ not computed
184.96.1.cn.1 $184$ $2$ $2$ $1$ $?$ dimension zero
208.96.3.hn.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.ho.1 $208$ $2$ $2$ $3$ $?$ not computed
208.96.3.hp.1 $208$ $2$ $2$ $3$ $?$ not computed
232.96.1.cn.1 $232$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.tj.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.tk.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.tl.1 $240$ $2$ $2$ $3$ $?$ not computed
248.96.1.cn.1 $248$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qs.1 $264$ $2$ $2$ $1$ $?$ dimension zero
272.96.3.hn.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.ho.1 $272$ $2$ $2$ $3$ $?$ not computed
272.96.3.hp.1 $272$ $2$ $2$ $3$ $?$ not computed
280.96.1.px.1 $280$ $2$ $2$ $1$ $?$ dimension zero
296.96.1.cn.1 $296$ $2$ $2$ $1$ $?$ dimension zero
304.96.3.gl.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.gm.1 $304$ $2$ $2$ $3$ $?$ not computed
304.96.3.gn.1 $304$ $2$ $2$ $3$ $?$ not computed
312.96.1.qs.1 $312$ $2$ $2$ $1$ $?$ dimension zero
328.96.1.cn.1 $328$ $2$ $2$ $1$ $?$ dimension zero